Collective elastic properties of mechanical metamaterials

Recent studies of the unusual frequency–wavenumber dependences and the related phenomena of the negative effective mass and stiffness in phononic metamaterials unveiled many futuristic opportunities for the acoustical and heat transfer engineering [1], [2], [3], [4], [5]. They also facilitated investigations of quasistatic responses of architected material systems in a quest for negative elastic moduli [6], [7], [8], [9], [10] and other unusual basic properties [11], [12], [13], [14], [15] expected at uniform loading conditions. In this letter, we show that materials with discrete internal structure may also demonstrate strong dependences of their effective (continuum equivalent) mechanical properties on the wavenumber/spatial frequency of static sinusoidal pressure waves. The Fourier mode stiffness, a ratio of the periodic load amplitude and a response amplitude is a quadratic function of the load's wavenumber in continuum materials. However, in discrete periodic lattices, we show it to be a squared sine function of the wavenumber, which flattens down when the wavelength becomes comparable with a unit cell size. As a result, comparing similarly loaded continuum and lattice materials leads to a varying dependence of the effective longitudinal and shear moduli on the wavenumber. Practically, this means that both material design and static load patterning can influence an effective elastic modulus of a mechanical metamaterial, and it can be seen from collective responses of a large group of unit cells.